Distance on a great circle

Percentage Accurate: 62.7% → 98.6%

Time: 34.7s

Alternatives: 35

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\\ t_1 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (-
          (* (sin (* lambda1 0.5)) (cos (* lambda2 0.5)))
          (* (cos (* lambda1 0.5)) (sin (* lambda2 0.5)))))
        (t_1
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* -0.5 phi1))
            (* (sin (* 0.5 phi1)) (cos (* phi2 0.5))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (sin((lambda1 * 0.5)) * cos((lambda2 * 0.5))) - (cos((lambda1 * 0.5)) * sin((lambda2 * 0.5)));
	double t_1 = pow(fma(sin((-0.5 * phi2)), cos((-0.5 * phi1)), (sin((0.5 * phi1)) * cos((phi2 * 0.5)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(Float64(lambda1 * 0.5)) * cos(Float64(lambda2 * 0.5))) - Float64(cos(Float64(lambda1 * 0.5)) * sin(Float64(lambda2 * 0.5))))
	t_1 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(-0.5 * phi1)), Float64(sin(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 63.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 78.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. mult-flip N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. mult-flip N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 78.0

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

7. Applied rewrites 78.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. mult-flip N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. mult-flip N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 79.5

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

9. Applied rewrites 79.5%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. mult-flip N/A

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

    10. metadata-eval N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cos.f64 N/A

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

    13. mult-flip N/A

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

    14. metadata-eval N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 N/A

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

    17. lower-cos.f64 N/A

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

    18. mult-flip N/A

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

    19. metadata-eval N/A

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

    20. lower-*.f64 N/A

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

    21. lower-sin.f64 N/A

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

    22. mult-flip N/A

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

    23. metadata-eval N/A

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

    24. lower-*.f64 79.2

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

11. Applied rewrites 79.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. mult-flip N/A

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

    10. metadata-eval N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cos.f64 N/A

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

    13. mult-flip N/A

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

    14. metadata-eval N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 N/A

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

    17. lower-cos.f64 N/A

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

    18. mult-flip N/A

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

    19. metadata-eval N/A

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

    20. lower-*.f64 N/A

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

    21. lower-sin.f64 N/A

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

    22. mult-flip N/A

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

    23. metadata-eval N/A

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

    24. lower-*.f64 98.6

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

13. Applied rewrites 98.6%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. mult-flip N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. *-commutative N/A

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

    16. distribute-lft-neg-in N/A

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

    17. metadata-eval N/A

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

    18. lower-*.f64 N/A

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

    19. lift-*.f64 N/A

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

    20. *-commutative N/A

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

    21. lift-*.f64 N/A

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

    22. lower-*.f64 98.6

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

    23. lift-cos.f64 N/A

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

15. Applied rewrites 98.6%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. mult-flip N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. *-commutative N/A

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

    16. distribute-lft-neg-in N/A

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

    17. metadata-eval N/A

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

    18. lower-*.f64 N/A

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

    19. lift-*.f64 N/A

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

    20. *-commutative N/A

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

    21. lift-*.f64 N/A

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

    22. lower-*.f64 98.6

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

    23. lift-cos.f64 N/A

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

17. Applied rewrites 98.6%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 63.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 78.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. mult-flip N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. mult-flip N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 78.0

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

7. Applied rewrites 78.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. mult-flip N/A

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

   19. metadata-eval N/A

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

   20. lower-*.f64 N/A

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

   21. lower-sin.f64 N/A

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

   22. mult-flip N/A

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

   23. metadata-eval N/A

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

   24. lower-*.f64 79.5

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

9. Applied rewrites 79.5%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. mult-flip N/A

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

    10. metadata-eval N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cos.f64 N/A

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

    13. mult-flip N/A

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

    14. metadata-eval N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 N/A

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

    17. lower-cos.f64 N/A

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

    18. mult-flip N/A

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

    19. metadata-eval N/A

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

    20. lower-*.f64 N/A

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

    21. lower-sin.f64 N/A

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

    22. mult-flip N/A

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

    23. metadata-eval N/A

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

    24. lower-*.f64 79.2

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

11. Applied rewrites 79.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. mult-flip N/A

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

    10. metadata-eval N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cos.f64 N/A

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

    13. mult-flip N/A

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

    14. metadata-eval N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 N/A

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

    17. lower-cos.f64 N/A

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

    18. mult-flip N/A

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

    19. metadata-eval N/A

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

    20. lower-*.f64 N/A

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

    21. lower-sin.f64 N/A

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

    22. mult-flip N/A

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

    23. metadata-eval N/A

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

    24. lower-*.f64 98.6

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

13. Applied rewrites 98.6%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. mult-flip N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. *-commutative N/A

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

    16. distribute-lft-neg-in N/A

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

    17. metadata-eval N/A

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

    18. lower-*.f64 N/A

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

    19. lift-*.f64 N/A

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

    20. *-commutative N/A

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

    21. lift-*.f64 N/A

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

    22. lower-*.f64 98.6

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

    23. lift-cos.f64 N/A

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

15. Applied rewrites 98.6%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. mult-flip N/A

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

    8. metadata-eval N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. *-commutative N/A

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

    16. distribute-lft-neg-in N/A

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

    17. metadata-eval N/A

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

    18. lower-*.f64 N/A

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

    19. lift-*.f64 N/A

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

    20. *-commutative N/A

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

    21. lift-*.f64 N/A

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

    22. lower-*.f64 98.6

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

    23. lift-cos.f64 N/A

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

17. Applied rewrites 98.6%

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

18. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

20. Applied rewrites 98.6%

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

21. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

23. Applied rewrites 98.6%

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

Alternative 3: 88.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(-0.5 \cdot \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \left(-0.5 \cdot \phi_1\right)\\ t_3 := \sin \left(0.5 \cdot \phi_1\right)\\ t_4 := \cos \left(\phi_2 \cdot 0.5\right)\\ t_5 := {\left(\mathsf{fma}\left(t\_0, t\_2, t\_3 \cdot t\_4\right)\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}\\ t_6 := \cos \phi_2 \cdot \cos \phi_1\\ t_7 := {\left(\mathsf{fma}\left(t\_0, t\_2, t\_4 \cdot t\_3\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_8 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(t\_6, 0.5, t\_6 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\\ \mathbf{if}\;\phi_2 \leq -0.00016:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_8}}{\sqrt{1 - t\_8}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_7}}{\sqrt{1 - t\_7}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* -0.5 phi2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (cos (* -0.5 phi1)))
        (t_3 (sin (* 0.5 phi1)))
        (t_4 (cos (* phi2 0.5)))
        (t_5
         (+
          (pow (fma t_0 t_2 (* t_3 t_4)) 2.0)
          (*
           (cos phi1)
           (pow
            (-
             (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
             (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
            2.0))))
        (t_6 (* (cos phi2) (cos phi1)))
        (t_7
         (+
          (pow (fma t_0 t_2 (* t_4 t_3)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (t_8
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (fma t_6 0.5 (* t_6 (* (cos (- lambda2 lambda1)) -0.5))))))
   (if (<= phi2 -0.00016)
     (* R (* 2.0 (atan2 (sqrt t_8) (sqrt (- 1.0 t_8)))))
     (if (<= phi2 3.4e-8)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (* R (* 2.0 (atan2 (sqrt t_7) (sqrt (- 1.0 t_7)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((-0.5 * phi2));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos((-0.5 * phi1));
	double t_3 = sin((0.5 * phi1));
	double t_4 = cos((phi2 * 0.5));
	double t_5 = pow(fma(t_0, t_2, (t_3 * t_4)), 2.0) + (cos(phi1) * pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0));
	double t_6 = cos(phi2) * cos(phi1);
	double t_7 = pow(fma(t_0, t_2, (t_4 * t_3)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double t_8 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + fma(t_6, 0.5, (t_6 * (cos((lambda2 - lambda1)) * -0.5)));
	double tmp;
	if (phi2 <= -0.00016) {
		tmp = R * (2.0 * atan2(sqrt(t_8), sqrt((1.0 - t_8))));
	} else if (phi2 <= 3.4e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_7), sqrt((1.0 - t_7))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = cos(Float64(-0.5 * phi1))
	t_3 = sin(Float64(0.5 * phi1))
	t_4 = cos(Float64(phi2 * 0.5))
	t_5 = Float64((fma(t_0, t_2, Float64(t_3 * t_4)) ^ 2.0) + Float64(cos(phi1) * (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0)))
	t_6 = Float64(cos(phi2) * cos(phi1))
	t_7 = Float64((fma(t_0, t_2, Float64(t_4 * t_3)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	t_8 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + fma(t_6, 0.5, Float64(t_6 * Float64(cos(Float64(lambda2 - lambda1)) * -0.5))))
	tmp = 0.0
	if (phi2 <= -0.00016)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_8), sqrt(Float64(1.0 - t_8)))));
	elseif (phi2 <= 3.4e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_7), sqrt(Float64(1.0 - t_7)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$0 * t$95$2 + N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[Power[N[(t$95$0 * t$95$2 + N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$6 * 0.5 + N[(t$95$6 * N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.00016], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$8], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$8), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.4e-8], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$7], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$7), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.60000000000000013e-4

   1. Initial program 62.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   5. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

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

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

      17. lift-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

   7. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

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

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

      17. lift-*.f64 N/A

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

      18. lift-cos.f64 N/A

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

   9. Applied rewrites 76.5%

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

   if -1.60000000000000013e-4 < phi2 < 3.4e-8

   1. Initial program 62.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   5. Applied rewrites 78.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

      22. mult-flip N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 78.0

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

   7. Applied rewrites 78.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. mult-flip N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. lower-sin.f64 N/A

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

      22. mult-flip N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 79.5

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

   9. Applied rewrites 79.5%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sin-diff N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. mult-flip N/A

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

       10. metadata-eval N/A

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

       11. lower-*.f64 N/A

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

       12. lower-cos.f64 N/A

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

       13. mult-flip N/A

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

       14. metadata-eval N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-cos.f64 N/A

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

       18. mult-flip N/A

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

       19. metadata-eval N/A

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

       20. lower-*.f64 N/A

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

       21. lower-sin.f64 N/A

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

       22. mult-flip N/A

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

       23. metadata-eval N/A

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

       24. lower-*.f64 79.2

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

   11. Applied rewrites 79.2%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sin-diff N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. mult-flip N/A

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

       10. metadata-eval N/A

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

       11. lower-*.f64 N/A

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

       12. lower-cos.f64 N/A

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

       13. mult-flip N/A

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

       14. metadata-eval N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-cos.f64 N/A

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

       18. mult-flip N/A

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

       19. metadata-eval N/A

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

       20. lower-*.f64 N/A

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

       21. lower-sin.f64 N/A

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

       22. mult-flip N/A

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

       23. metadata-eval N/A

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

       24. lower-*.f64 98.6

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

   13. Applied rewrites 98.6%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. mult-flip N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-cos.f64 N/A

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

       12. cos-neg-rev N/A

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

       13. lower-cos.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. distribute-lft-neg-in N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lift-*.f64 N/A

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

       22. lower-*.f64 98.6

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

       23. lift-cos.f64 N/A

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

   15. Applied rewrites 98.6%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. mult-flip N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lift-cos.f64 N/A

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

       12. cos-neg-rev N/A

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

       13. lower-cos.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. *-commutative N/A

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

       16. distribute-lft-neg-in N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lift-*.f64 N/A

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

       22. lower-*.f64 98.6

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

       23. lift-cos.f64 N/A

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

   17. Applied rewrites 98.6%

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

   18. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-pow.f64 N/A

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

   20. Applied rewrites 69.6%

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

   21. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-pow.f64 N/A

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

   23. Applied rewrites 66.2%

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

   if 3.4e-8 < phi2

   1. Initial program 62.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   5. Applied rewrites 78.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\phi_1 \cdot \frac{1}{2}}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\phi_1 \cdot \frac{1}{2}}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \left(\phi_2 - \phi_1\right)\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, \mathsf{fma}\left(-0.5, t\_0, 0.5\right), 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_3, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ t_5 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;\lambda_1 \leq -1.25 \cdot 10^{+26}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\lambda_1 \leq 1.26 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (cos (- phi2 phi1)))
        (t_3 (* (cos phi2) (cos phi1)))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_3 (fma -0.5 t_0 0.5) (- 0.5 (* t_2 0.5))))
            (sqrt (fma (fma t_0 0.5 -0.5) t_3 (fma t_2 0.5 0.5))))
           2.0)
          R))
        (t_5
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* -0.5 phi1))
            (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
   (if (<= lambda1 -1.25e+26)
     t_4
     (if (<= lambda1 1.26e-5)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos((phi2 - phi1));
	double t_3 = cos(phi2) * cos(phi1);
	double t_4 = (atan2(sqrt(fma(t_3, fma(-0.5, t_0, 0.5), (0.5 - (t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	double t_5 = pow(fma(sin((-0.5 * phi2)), cos((-0.5 * phi1)), (cos((phi2 * 0.5)) * sin((0.5 * phi1)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double tmp;
	if (lambda1 <= -1.25e+26) {
		tmp = t_4;
	} else if (lambda1 <= 1.26e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = cos(Float64(phi2 - phi1))
	t_3 = Float64(cos(phi2) * cos(phi1))
	t_4 = Float64(Float64(atan(sqrt(fma(t_3, fma(-0.5, t_0, 0.5), Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R)
	t_5 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(-0.5 * phi1)), Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	tmp = 0.0
	if (lambda1 <= -1.25e+26)
		tmp = t_4;
	elseif (lambda1 <= 1.26e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$3 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.25e+26], t$95$4, If[LessEqual[lambda1, 1.26e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.25e26 or 1.25999999999999996e-5 < lambda1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.3

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.3%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 72.5

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 72.5%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -1.25e26 < lambda1 < 1.25999999999999996e-5

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\phi_1 \cdot \frac{1}{2}}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)} \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\phi_1 \cdot \frac{1}{2}}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. distribute-lft-neg-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := \cos \left(\phi_2 - \phi_1\right)\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(-0.5, t\_0, 0.5\right), 0.5 - t\_3 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_2, \mathsf{fma}\left(t\_3, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\lambda_1 \leq 4.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (cos (- phi2 phi1)))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_2 (fma -0.5 t_0 0.5) (- 0.5 (* t_3 0.5))))
            (sqrt (fma (fma t_0 0.5 -0.5) t_2 (fma t_3 0.5 0.5))))
           2.0)
          R)))
   (if (<= lambda1 -4.6e-7)
     t_4
     (if (<= lambda1 4.7e-6)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow
             (fma
              (sin (* phi1 0.5))
              (cos (/ phi2 -2.0))
              (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
             2.0)
            (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (cos phi2) (pow (sin (* -0.5 lambda2)) 2.0))
             (pow
              (fma
               (cos (* -0.5 phi2))
               (sin (* 0.5 phi1))
               (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
              2.0)))))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = cos((phi2 - phi1));
	double t_4 = (atan2(sqrt(fma(t_2, fma(-0.5, t_0, 0.5), (0.5 - (t_3 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_2, fma(t_3, 0.5, 0.5)))) * 2.0) * R;
	double tmp;
	if (lambda1 <= -4.6e-7) {
		tmp = t_4;
	} else if (lambda1 <= 4.7e-6) {
		tmp = R * (2.0 * atan2(sqrt((pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * pow(sin((-0.5 * lambda2)), 2.0)), pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0))))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = cos(Float64(phi2 - phi1))
	t_4 = Float64(Float64(atan(sqrt(fma(t_2, fma(-0.5, t_0, 0.5), Float64(0.5 - Float64(t_3 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_2, fma(t_3, 0.5, 0.5)))) * 2.0) * R)
	tmp = 0.0
	if (lambda1 <= -4.6e-7)
		tmp = t_4;
	elseif (lambda1 <= 4.7e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * (sin(Float64(-0.5 * lambda2)) ^ 2.0)), (fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0)))))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$2 * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$2 + N[(t$95$3 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[lambda1, -4.6e-7], t$95$4, If[LessEqual[lambda1, 4.7e-6], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.5999999999999999e-7 or 4.69999999999999989e-6 < lambda1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.3

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.3%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 72.5

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 72.5%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -4.5999999999999999e-7 < lambda1 < 4.69999999999999989e-6

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   8. Applied rewrites 59.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := \cos \left(\phi_2 - \phi_1\right)\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(-0.5, t\_1, 0.5\right), 0.5 - t\_3 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 0.5, -0.5\right), t\_2, \mathsf{fma}\left(t\_3, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\lambda_1 \leq 4.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, t\_0\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_1 (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (cos (- phi2 phi1)))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_2 (fma -0.5 t_1 0.5) (- 0.5 (* t_3 0.5))))
            (sqrt (fma (fma t_1 0.5 -0.5) t_2 (fma t_3 0.5 0.5))))
           2.0)
          R)))
   (if (<= lambda1 -4.6e-7)
     t_4
     (if (<= lambda1 4.7e-6)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (* (cos phi2) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
            t_0))
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (cos phi2) (pow (sin (* -0.5 lambda2)) 2.0))
             t_0))))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_1 = fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = cos((phi2 - phi1));
	double t_4 = (atan2(sqrt(fma(t_2, fma(-0.5, t_1, 0.5), (0.5 - (t_3 * 0.5)))), sqrt(fma(fma(t_1, 0.5, -0.5), t_2, fma(t_3, 0.5, 0.5)))) * 2.0) * R;
	double tmp;
	if (lambda1 <= -4.6e-7) {
		tmp = t_4;
	} else if (lambda1 <= 4.7e-6) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)), t_0)), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * pow(sin((-0.5 * lambda2)), 2.0)), t_0)))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_1 = fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1)))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = cos(Float64(phi2 - phi1))
	t_4 = Float64(Float64(atan(sqrt(fma(t_2, fma(-0.5, t_1, 0.5), Float64(0.5 - Float64(t_3 * 0.5)))), sqrt(fma(fma(t_1, 0.5, -0.5), t_2, fma(t_3, 0.5, 0.5)))) * 2.0) * R)
	tmp = 0.0
	if (lambda1 <= -4.6e-7)
		tmp = t_4;
	elseif (lambda1 <= 4.7e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)), t_0)), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * (sin(Float64(-0.5 * lambda2)) ^ 2.0)), t_0))))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$2 * N[(-0.5 * t$95$1 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * 0.5 + -0.5), $MachinePrecision] * t$95$2 + N[(t$95$3 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[lambda1, -4.6e-7], t$95$4, If[LessEqual[lambda1, 4.7e-6], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.5999999999999999e-7 or 4.69999999999999989e-6 < lambda1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.3

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.3%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 72.5

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 72.5%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -4.5999999999999999e-7 < lambda1 < 4.69999999999999989e-6

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 59.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_1}^{2}\right)\\ t_4 := \cos \phi_2 \cdot \cos \phi_1\\ t_5 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), t\_1 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + t\_4 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot 0.5 - \lambda_2 \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -0.00011:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4, \mathsf{fma}\left(-0.5, t\_0, 0.5\right), 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_4, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1))))
        (t_3
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_1 2.0)))
        (t_4 (* (cos phi2) (cos phi1)))
        (t_5
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* -0.5 phi1))
            (* t_1 (cos (* phi2 0.5))))
           2.0)
          (*
           t_4
           (-
            0.5
            (* 0.5 (cos (* 2.0 (- (* lambda1 0.5) (* lambda2 0.5))))))))))
   (if (<= phi2 -0.00011)
     (*
      (*
       (atan2
        (sqrt (fma t_4 (fma -0.5 t_0 0.5) (- 0.5 (* t_2 0.5))))
        (sqrt (fma (fma t_0 0.5 -0.5) t_4 (fma t_2 0.5 0.5))))
       2.0)
      R)
     (if (<= phi2 5.5e-31)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin((0.5 * phi1));
	double t_2 = fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1)));
	double t_3 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_1, 2.0));
	double t_4 = cos(phi2) * cos(phi1);
	double t_5 = pow(fma(sin((-0.5 * phi2)), cos((-0.5 * phi1)), (t_1 * cos((phi2 * 0.5)))), 2.0) + (t_4 * (0.5 - (0.5 * cos((2.0 * ((lambda1 * 0.5) - (lambda2 * 0.5)))))));
	double tmp;
	if (phi2 <= -0.00011) {
		tmp = (atan2(sqrt(fma(t_4, fma(-0.5, t_0, 0.5), (0.5 - (t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_4, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	} else if (phi2 <= 5.5e-31) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1)))
	t_3 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_1 ^ 2.0))
	t_4 = Float64(cos(phi2) * cos(phi1))
	t_5 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(-0.5 * phi1)), Float64(t_1 * cos(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(t_4 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 * 0.5) - Float64(lambda2 * 0.5))))))))
	tmp = 0.0
	if (phi2 <= -0.00011)
		tmp = Float64(Float64(atan(sqrt(fma(t_4, fma(-0.5, t_0, 0.5), Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_4, fma(t_2, 0.5, 0.5)))) * 2.0) * R);
	elseif (phi2 <= 5.5e-31)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$4 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 * 0.5), $MachinePrecision] - N[(lambda2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.00011], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$4 * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$4 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 5.5e-31], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.10000000000000004e-4

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.8

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.8%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 74.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 74.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -1.10000000000000004e-4 < phi2 < 5.49999999999999958e-31

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 78.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 79.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 79.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 5.49999999999999958e-31 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 78.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 79.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 79.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\phi_1 \cdot \frac{1}{2}}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       16. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       17. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       21. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \color{blue}{\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

       23. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \color{blue}{\cos \left(\mathsf{neg}\left(\phi_1 \cdot \frac{1}{2}\right)\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\phi_1 \cdot \frac{1}{2}}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       16. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       17. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \color{blue}{\left(\frac{-1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       21. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \color{blue}{\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

       23. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)} \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. associate-*l* N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       7. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{\left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   19. Applied rewrites 77.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot 0.5 - \lambda_2 \cdot 0.5\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]
   20. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot \frac{1}{2} - \lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot \frac{1}{2} - \lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)} \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. associate-*l* N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot \frac{1}{2} - \lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot \frac{1}{2} - \lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot \frac{1}{2} - \lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot \frac{1}{2} - \lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)\right)}}\right) \]

       7. lower-*.f32 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot \frac{1}{2} - \lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\frac{-1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{\left(\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}\right)}}\right) \]

   21. Applied rewrites 76.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot 0.5 - \lambda_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_1 \cdot 0.5 - \lambda_2 \cdot 0.5\right)\right)\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \mathsf{fma}\left(-0.5, t\_0, 0.5\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_4 := \cos \phi_2 \cdot \cos \phi_1\\ t_5 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + t\_1 \cdot t\_4\\ \mathbf{if}\;\phi_2 \leq -0.00011:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4, t\_1, 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_4, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (fma -0.5 t_0 0.5))
        (t_2 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1))))
        (t_3
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_4 (* (cos phi2) (cos phi1)))
        (t_5
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (* t_1 t_4))))
   (if (<= phi2 -0.00011)
     (*
      (*
       (atan2
        (sqrt (fma t_4 t_1 (- 0.5 (* t_2 0.5))))
        (sqrt (fma (fma t_0 0.5 -0.5) t_4 (fma t_2 0.5 0.5))))
       2.0)
      R)
     (if (<= phi2 5.5e-31)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = fma(-0.5, t_0, 0.5);
	double t_2 = fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1)));
	double t_3 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double t_4 = cos(phi2) * cos(phi1);
	double t_5 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (t_1 * t_4);
	double tmp;
	if (phi2 <= -0.00011) {
		tmp = (atan2(sqrt(fma(t_4, t_1, (0.5 - (t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_4, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	} else if (phi2 <= 5.5e-31) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = fma(-0.5, t_0, 0.5)
	t_2 = fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1)))
	t_3 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_4 = Float64(cos(phi2) * cos(phi1))
	t_5 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(t_1 * t_4))
	tmp = 0.0
	if (phi2 <= -0.00011)
		tmp = Float64(Float64(atan(sqrt(fma(t_4, t_1, Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_4, fma(t_2, 0.5, 0.5)))) * 2.0) * R);
	elseif (phi2 <= 5.5e-31)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * t$95$0 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.00011], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$4 * t$95$1 + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$4 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 5.5e-31], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.10000000000000004e-4

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.8

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.8%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 74.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 74.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -1.10000000000000004e-4 < phi2 < 5.49999999999999958e-31

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 78.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 79.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 79.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 5.49999999999999958e-31 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}\right)}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)\right)}}\right) \]

      8. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \mathsf{fma}\left(t\_1, 0.5, -0.5\right)\\ t_3 := \mathsf{fma}\left(-0.5, t\_1, 0.5\right)\\ t_4 := \sin \phi_2 \cdot \sin \phi_1\\ t_5 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, t\_4\right)\\ t_6 := t\_0 + t\_4\\ t_7 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -0.00011:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, t\_3, 0.5 - t\_5 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_2, t\_0, \mathsf{fma}\left(t\_5, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_7}}{\sqrt{1 - t\_7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, t\_3, 0.5 - t\_6 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_2, t\_0, \mathsf{fma}\left(t\_6, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2 (fma t_1 0.5 -0.5))
        (t_3 (fma -0.5 t_1 0.5))
        (t_4 (* (sin phi2) (sin phi1)))
        (t_5 (fma (cos phi2) (cos phi1) t_4))
        (t_6 (+ t_0 t_4))
        (t_7
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow (sin (* 0.5 phi1)) 2.0))))
   (if (<= phi2 -0.00011)
     (*
      (*
       (atan2
        (sqrt (fma t_0 t_3 (- 0.5 (* t_5 0.5))))
        (sqrt (fma t_2 t_0 (fma t_5 0.5 0.5))))
       2.0)
      R)
     (if (<= phi2 9.2e-10)
       (* R (* 2.0 (atan2 (sqrt t_7) (sqrt (- 1.0 t_7)))))
       (*
        (*
         (atan2
          (sqrt (fma t_0 t_3 (- 0.5 (* t_6 0.5))))
          (sqrt (fma t_2 t_0 (fma t_6 0.5 0.5))))
         2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = fma(t_1, 0.5, -0.5);
	double t_3 = fma(-0.5, t_1, 0.5);
	double t_4 = sin(phi2) * sin(phi1);
	double t_5 = fma(cos(phi2), cos(phi1), t_4);
	double t_6 = t_0 + t_4;
	double t_7 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if (phi2 <= -0.00011) {
		tmp = (atan2(sqrt(fma(t_0, t_3, (0.5 - (t_5 * 0.5)))), sqrt(fma(t_2, t_0, fma(t_5, 0.5, 0.5)))) * 2.0) * R;
	} else if (phi2 <= 9.2e-10) {
		tmp = R * (2.0 * atan2(sqrt(t_7), sqrt((1.0 - t_7))));
	} else {
		tmp = (atan2(sqrt(fma(t_0, t_3, (0.5 - (t_6 * 0.5)))), sqrt(fma(t_2, t_0, fma(t_6, 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = fma(t_1, 0.5, -0.5)
	t_3 = fma(-0.5, t_1, 0.5)
	t_4 = Float64(sin(phi2) * sin(phi1))
	t_5 = fma(cos(phi2), cos(phi1), t_4)
	t_6 = Float64(t_0 + t_4)
	t_7 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -0.00011)
		tmp = Float64(Float64(atan(sqrt(fma(t_0, t_3, Float64(0.5 - Float64(t_5 * 0.5)))), sqrt(fma(t_2, t_0, fma(t_5, 0.5, 0.5)))) * 2.0) * R);
	elseif (phi2 <= 9.2e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_7), sqrt(Float64(1.0 - t_7)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_0, t_3, Float64(0.5 - Float64(t_6 * 0.5)))), sqrt(fma(t_2, t_0, fma(t_6, 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * t$95$1 + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$0 + t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.00011], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$0 * t$95$3 + N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$0 + N[(t$95$5 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 9.2e-10], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$7], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$7), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$0 * t$95$3 + N[(0.5 - N[(t$95$6 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$0 + N[(t$95$6 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.10000000000000004e-4

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.8

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.8%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 74.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 74.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -1.10000000000000004e-4 < phi2 < 9.20000000000000028e-10

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 78.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 79.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 79.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 9.20000000000000028e-10 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      10. lower-sin.f64 58.8

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.8%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       10. lower-sin.f64 74.2

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 74.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, \mathsf{fma}\left(-0.5, t\_0, 0.5\right), 0.5 - t\_1 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_3, \mathsf{fma}\left(t\_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.00011:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1))))
        (t_2
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (* (cos phi2) (cos phi1)))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_3 (fma -0.5 t_0 0.5) (- 0.5 (* t_1 0.5))))
            (sqrt (fma (fma t_0 0.5 -0.5) t_3 (fma t_1 0.5 0.5))))
           2.0)
          R)))
   (if (<= phi2 -0.00011)
     t_4
     (if (<= phi2 9.2e-10)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1)));
	double t_2 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double t_3 = cos(phi2) * cos(phi1);
	double t_4 = (atan2(sqrt(fma(t_3, fma(-0.5, t_0, 0.5), (0.5 - (t_1 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_1, 0.5, 0.5)))) * 2.0) * R;
	double tmp;
	if (phi2 <= -0.00011) {
		tmp = t_4;
	} else if (phi2 <= 9.2e-10) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1)))
	t_2 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = Float64(cos(phi2) * cos(phi1))
	t_4 = Float64(Float64(atan(sqrt(fma(t_3, fma(-0.5, t_0, 0.5), Float64(0.5 - Float64(t_1 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_1, 0.5, 0.5)))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -0.00011)
		tmp = t_4;
	elseif (phi2 <= 9.2e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$3 + N[(t$95$1 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.00011], t$95$4, If[LessEqual[phi2, 9.2e-10], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.10000000000000004e-4 or 9.20000000000000028e-10 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.8

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.8%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 74.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 74.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -1.10000000000000004e-4 < phi2 < 9.20000000000000028e-10

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 78.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 79.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 79.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \left(\phi_2 - \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, \mathsf{fma}\left(-0.5, t\_0, 0.5\right), 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_3, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ t_5 := \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -0.00016:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, \mathsf{fma}\left(-0.5, t\_5, 0.5\right), 0.5 - t\_1 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_5, 0.5, -0.5\right), t\_3, \mathsf{fma}\left(t\_1, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (cos (- phi2 phi1)))
        (t_2 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1))))
        (t_3 (* (cos phi2) (cos phi1)))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_3 (fma -0.5 t_0 0.5) (- 0.5 (* t_2 0.5))))
            (sqrt (fma (fma t_0 0.5 -0.5) t_3 (fma t_2 0.5 0.5))))
           2.0)
          R))
        (t_5
         (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))
   (if (<= phi2 -0.00016)
     t_4
     (if (<= phi2 3.5e-12)
       (*
        (*
         (atan2
          (sqrt (fma t_3 (fma -0.5 t_5 0.5) (- 0.5 (* t_1 0.5))))
          (sqrt (fma (fma t_5 0.5 -0.5) t_3 (fma t_1 0.5 0.5))))
         2.0)
        R)
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = cos((phi2 - phi1));
	double t_2 = fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1)));
	double t_3 = cos(phi2) * cos(phi1);
	double t_4 = (atan2(sqrt(fma(t_3, fma(-0.5, t_0, 0.5), (0.5 - (t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	double t_5 = fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)));
	double tmp;
	if (phi2 <= -0.00016) {
		tmp = t_4;
	} else if (phi2 <= 3.5e-12) {
		tmp = (atan2(sqrt(fma(t_3, fma(-0.5, t_5, 0.5), (0.5 - (t_1 * 0.5)))), sqrt(fma(fma(t_5, 0.5, -0.5), t_3, fma(t_1, 0.5, 0.5)))) * 2.0) * R;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = cos(Float64(phi2 - phi1))
	t_2 = fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1)))
	t_3 = Float64(cos(phi2) * cos(phi1))
	t_4 = Float64(Float64(atan(sqrt(fma(t_3, fma(-0.5, t_0, 0.5), Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R)
	t_5 = fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1)))
	tmp = 0.0
	if (phi2 <= -0.00016)
		tmp = t_4;
	elseif (phi2 <= 3.5e-12)
		tmp = Float64(Float64(atan(sqrt(fma(t_3, fma(-0.5, t_5, 0.5), Float64(0.5 - Float64(t_1 * 0.5)))), sqrt(fma(fma(t_5, 0.5, -0.5), t_3, fma(t_1, 0.5, 0.5)))) * 2.0) * R);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$3 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.00016], t$95$4, If[LessEqual[phi2, 3.5e-12], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * N[(-0.5 * t$95$5 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$5 * 0.5 + -0.5), $MachinePrecision] * t$95$3 + N[(t$95$1 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.60000000000000013e-4 or 3.5e-12 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.8

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.8%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 74.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 74.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -1.60000000000000013e-4 < phi2 < 3.5e-12

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.3

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.3%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 72.5

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 72.5%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}, 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_5 := t\_3 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_6 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 0.03:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.5, t\_6, 0.5\right), 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_6, 0.5, -0.5\right), t\_0, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1))))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (t_5
         (+ t_3 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_6 (cos (- lambda2 lambda1))))
   (if (<= (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 0.03)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (*
      (*
       (atan2
        (sqrt (fma t_0 (fma -0.5 t_6 0.5) (- 0.5 (* t_2 0.5))))
        (sqrt (fma (fma t_6 0.5 -0.5) t_0 (fma t_2 0.5 0.5))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1)));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = t_3 + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double t_5 = t_3 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_6 = cos((lambda2 - lambda1));
	double tmp;
	if ((2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) <= 0.03) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = (atan2(sqrt(fma(t_0, fma(-0.5, t_6, 0.5), (0.5 - (t_2 * 0.5)))), sqrt(fma(fma(t_6, 0.5, -0.5), t_0, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1)))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	t_5 = Float64(t_3 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_6 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) <= 0.03)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_0, fma(-0.5, t_6, 0.5), Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(fma(fma(t_6, 0.5, -0.5), t_0, fma(t_2, 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.03], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$0 * N[(-0.5 * t$95$6 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$6 * 0.5 + -0.5), $MachinePrecision] * t$95$0 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 0.029999999999999999

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.029999999999999999 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lift--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. cos-diff N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-sin.f64 58.8

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 58.8%

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 74.2

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   11. Applied rewrites 74.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\lambda_1 - \lambda_2}{2}\\ t_1 := \sin t\_0\\ t_2 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_3 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + t\_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right) \cdot \cos \phi_1, \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_2}}{\sqrt{\left(0.5 + 0.5 \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (/ (- lambda1 lambda2) 2.0))
        (t_1 (sin t_0))
        (t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1))
        (t_3
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0)))
   (if (<= t_0 -1e-21)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          t_2))
        (sqrt
         (fma
          (* (fma (cos (- lambda2 lambda1)) 0.5 -0.5) (cos phi1))
          (cos phi2)
          (fma (cos (- phi2 phi1)) 0.5 0.5))))))
     (if (<= t_0 2e-23)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_2))
          (sqrt
           (-
            (+
             0.5
             (* 0.5 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1)))))
            (*
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
             (* (cos phi2) (cos phi1))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) / 2.0;
	double t_1 = sin(t_0);
	double t_2 = ((cos(phi1) * cos(phi2)) * t_1) * t_1;
	double t_3 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double tmp;
	if (t_0 <= -1e-21) {
		tmp = R * (2.0 * atan2(sqrt((pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + t_2)), sqrt(fma((fma(cos((lambda2 - lambda1)), 0.5, -0.5) * cos(phi1)), cos(phi2), fma(cos((phi2 - phi1)), 0.5, 0.5)))));
	} else if (t_0 <= 2e-23) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_2)), sqrt(((0.5 + (0.5 * fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_1 = sin(t_0)
	t_2 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)
	t_3 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (t_0 <= -1e-21)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + t_2)), sqrt(fma(Float64(fma(cos(Float64(lambda2 - lambda1)), 0.5, -0.5) * cos(phi1)), cos(phi2), fma(cos(Float64(phi2 - phi1)), 0.5, 0.5))))));
	elseif (t_0 <= 2e-23)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_2)), sqrt(Float64(Float64(0.5 + Float64(0.5 * fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, -1e-21], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-23], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -9.99999999999999908e-22

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 63.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right) \cdot \cos \phi_1, \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}}}\right) \]

   if -9.99999999999999908e-22 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 1.99999999999999992e-23

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. lower-*.f64 35.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 35.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 35.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 1.99999999999999992e-23 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      4. associate-*r* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(2 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      5. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\color{blue}{1} \cdot \left(\phi_1 - \phi_2\right)\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      6. *-lft-identity N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\phi_1 - \phi_2\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      7. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\phi_1 - \phi_2\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      8. sub-negate-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\phi_2 - \phi_1\right)\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      9. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(\phi_2 - \phi_1\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      10. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      16. lower-sin.f64 63.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

   5. Applied rewrites 63.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_3 := t\_0 + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot t\_2}{2}\\ t_4 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ \mathbf{if}\;t\_1 \leq -0.0002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}}{\sqrt{\left(0.5 + 0.5 \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\right) - t\_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_3
         (+ t_0 (/ (* (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1))) t_2) 2.0)))
        (t_4
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0)))
   (if (<= t_1 -0.0002)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= t_1 2e-23)
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
          (sqrt
           (-
            (+
             0.5
             (* 0.5 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1)))))
            (* t_2 (* (cos phi2) (cos phi1))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_3 = t_0 + (((cos((phi2 - phi1)) + cos((phi2 + phi1))) * t_2) / 2.0);
	double t_4 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double tmp;
	if (t_1 <= -0.0002) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (t_1 <= 2e-23) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(((0.5 + (0.5 * fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1))))) - (t_2 * (cos(phi2) * cos(phi1)))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_3 = Float64(t_0 + Float64(Float64(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))) * t_2) / 2.0))
	t_4 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (t_1 <= -0.0002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (t_1 <= 2e-23)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(Float64(Float64(0.5 + Float64(0.5 * fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1))))) - Float64(t_2 * Float64(cos(phi2) * cos(phi1))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, -0.0002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-23], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -2.0000000000000001e-4

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   if -2.0000000000000001e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 1.99999999999999992e-23

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. lower-*.f64 35.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 35.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 35.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 1.99999999999999992e-23 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      4. associate-*r* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(2 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      5. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\color{blue}{1} \cdot \left(\phi_1 - \phi_2\right)\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      6. *-lft-identity N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\phi_1 - \phi_2\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      7. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\phi_1 - \phi_2\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      8. sub-negate-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\left(\phi_2 - \phi_1\right)\right)\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      9. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(\phi_2 - \phi_1\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      10. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(\frac{1}{2} + \frac{1}{2} \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

      16. lower-sin.f64 63.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]

   5. Applied rewrites 63.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(0.5 + 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 66.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \frac{\lambda_1 - \lambda_2}{2}\\ t_2 := \sin t\_1\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_4 := t\_0 + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot t\_3}{2}\\ t_5 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ \mathbf{if}\;t\_1 \leq -0.0002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2}}{\sqrt{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - t\_3 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right|}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (/ (- lambda1 lambda2) 2.0))
        (t_2 (sin t_1))
        (t_3 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_4
         (+ t_0 (/ (* (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1))) t_3) 2.0)))
        (t_5
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0)))
   (if (<= t_1 -0.0002)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= t_1 2e-23)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)))
          (sqrt
           (fabs
            (-
             (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
             (* t_3 (* (cos phi2) (cos phi1)))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = (lambda1 - lambda2) / 2.0;
	double t_2 = sin(t_1);
	double t_3 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_4 = t_0 + (((cos((phi2 - phi1)) + cos((phi2 + phi1))) * t_3) / 2.0);
	double t_5 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double tmp;
	if (t_1 <= -0.0002) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (t_1 <= 2e-23) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(fabs(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - (t_3 * (cos(phi2) * cos(phi1))))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_2 = sin(t_1)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_4 = Float64(t_0 + Float64(Float64(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))) * t_3) / 2.0))
	t_5 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (t_1 <= -0.0002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (t_1 <= 2e-23)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(abs(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(t_3 * Float64(cos(phi2) * cos(phi1)))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, -0.0002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-23], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -2.0000000000000001e-4

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   if -2.0000000000000001e-4 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 1.99999999999999992e-23

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. lower-*.f64 35.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 35.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 35.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 1.99999999999999992e-23 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right|}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \frac{\lambda_1 - \lambda_2}{2}\\ t_2 := \sin t\_1\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_4 := t\_0 + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot t\_3}{2}\\ t_5 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ \mathbf{if}\;t\_1 \leq -0.0002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - t\_3 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (/ (- lambda1 lambda2) 2.0))
        (t_2 (sin t_1))
        (t_3 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_4
         (+ t_0 (/ (* (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1))) t_3) 2.0)))
        (t_5
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0)))
   (if (<= t_1 -0.0002)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= t_1 2e-23)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)))
          (sqrt
           (-
            (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
            (* t_3 (* (cos phi2) (cos phi1))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = (lambda1 - lambda2) / 2.0;
	double t_2 = sin(t_1);
	double t_3 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_4 = t_0 + (((cos((phi2 - phi1)) + cos((phi2 + phi1))) * t_3) / 2.0);
	double t_5 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double tmp;
	if (t_1 <= -0.0002) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (t_1 <= 2e-23) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - (t_3 * (cos(phi2) * cos(phi1)))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_2 = sin(t_1)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_4 = Float64(t_0 + Float64(Float64(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))) * t_3) / 2.0))
	t_5 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (t_1 <= -0.0002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (t_1 <= 2e-23)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(t_3 * Float64(cos(phi2) * cos(phi1))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, -0.0002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-23], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -2.0000000000000001e-4

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   if -2.0000000000000001e-4 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 1.99999999999999992e-23

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. lower-*.f64 35.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 35.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 35.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 1.99999999999999992e-23 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;t\_2 \leq -0.0002:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (/
           (*
            (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1)))
            (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
           2.0)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= t_2 -0.0002)
     t_3
     (if (<= t_2 2e-23)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos((phi2 - phi1)) + cos((phi2 + phi1))) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))))) / 2.0);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (t_2 <= -0.0002) {
		tmp = t_3;
	} else if (t_2 <= 2e-23) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))) / 2.0))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (t_2 <= -0.0002)
		tmp = t_3;
	elseif (t_2 <= 2e-23)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.0002], t$95$3, If[LessEqual[t$95$2, 2e-23], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -2.0000000000000001e-4 or 1.99999999999999992e-23 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 61.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   if -2.0000000000000001e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 1.99999999999999992e-23

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. lower-*.f64 35.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 35.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 35.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_1 := \left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;t\_2 \leq -0.0002:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_1
         (fabs
          (fma
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
           (* (cos phi2) (cos phi1))
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= t_2 -0.0002)
     t_3
     (if (<= t_2 5e-20)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_1 = fabs(fma((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), (cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2))))))));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (t_2 <= -0.0002) {
		tmp = t_3;
	} else if (t_2 <= 5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_1 = abs(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (t_2 <= -0.0002)
		tmp = t_3;
	elseif (t_2 <= 5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.0002], t$95$3, If[LessEqual[t$95$2, 5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -2.0000000000000001e-4 or 4.9999999999999999e-20 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 58.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|}}{\sqrt{1 - \color{blue}{\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right|}}}\right) \]

   if -2.0000000000000001e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 4.9999999999999999e-20

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. lower-*.f64 35.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 35.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 35.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := \cos \left(\phi_2 - \phi_1\right)\\ t_3 := 0.5 - t\_2 \cdot 0.5\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_6 := \mathsf{fma}\left(t\_5, 0.5, -0.5\right)\\ \mathbf{if}\;t\_4 \leq -0.0002:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - t\_5 \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, t\_3\right)}}{\sqrt{\mathsf{fma}\left(t\_2 + 1, 0.5, t\_6 \cdot t\_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.5, t\_5, 0.5\right), t\_3\right)}}{\sqrt{\mathsf{fma}\left(t\_6, t\_1, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 phi2))
           (sin (* 0.5 phi1))
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_1 (* (cos phi2) (cos phi1)))
        (t_2 (cos (- phi2 phi1)))
        (t_3 (- 0.5 (* t_2 0.5)))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (cos (- lambda2 lambda1)))
        (t_6 (fma t_5 0.5 -0.5)))
   (if (<= t_4 -0.0002)
     (*
      (*
       (atan2
        (sqrt (fma (* (- 0.5 (* t_5 0.5)) (cos phi2)) (cos phi1) t_3))
        (sqrt (fma (+ t_2 1.0) 0.5 (* t_6 t_1))))
       2.0)
      R)
     (if (<= t_4 5e-10)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       (*
        (*
         (atan2
          (sqrt (fma t_1 (fma -0.5 t_5 0.5) t_3))
          (sqrt (fma t_6 t_1 (fma t_2 0.5 0.5))))
         2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_1 = cos(phi2) * cos(phi1);
	double t_2 = cos((phi2 - phi1));
	double t_3 = 0.5 - (t_2 * 0.5);
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = cos((lambda2 - lambda1));
	double t_6 = fma(t_5, 0.5, -0.5);
	double tmp;
	if (t_4 <= -0.0002) {
		tmp = (atan2(sqrt(fma(((0.5 - (t_5 * 0.5)) * cos(phi2)), cos(phi1), t_3)), sqrt(fma((t_2 + 1.0), 0.5, (t_6 * t_1)))) * 2.0) * R;
	} else if (t_4 <= 5e-10) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = (atan2(sqrt(fma(t_1, fma(-0.5, t_5, 0.5), t_3)), sqrt(fma(t_6, t_1, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_1 = Float64(cos(phi2) * cos(phi1))
	t_2 = cos(Float64(phi2 - phi1))
	t_3 = Float64(0.5 - Float64(t_2 * 0.5))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = cos(Float64(lambda2 - lambda1))
	t_6 = fma(t_5, 0.5, -0.5)
	tmp = 0.0
	if (t_4 <= -0.0002)
		tmp = Float64(Float64(atan(sqrt(fma(Float64(Float64(0.5 - Float64(t_5 * 0.5)) * cos(phi2)), cos(phi1), t_3)), sqrt(fma(Float64(t_2 + 1.0), 0.5, Float64(t_6 * t_1)))) * 2.0) * R);
	elseif (t_4 <= 5e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_1, fma(-0.5, t_5, 0.5), t_3)), sqrt(fma(t_6, t_1, fma(t_2, 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[t$95$4, -0.0002], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$2 + 1.0), $MachinePrecision] * 0.5 + N[(t$95$6 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[t$95$4, 5e-10], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$1 * N[(-0.5 * t$95$5 + 0.5), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$6 * t$95$1 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -2.0000000000000001e-4

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 78.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 79.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 79.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   15. Applied rewrites 57.9%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + 1, 0.5, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)}} \cdot 2\right) \cdot R} \]

   if -2.0000000000000001e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 5.00000000000000031e-10

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. lower-*.f64 35.6

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 35.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 35.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 5.00000000000000031e-10 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -3.35 \cdot 10^{-6}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 1.02 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_0)))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -3.35e-6)
     t_3
     (if (<= phi2 1.02e+27)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_0);
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -3.35e-6) {
		tmp = t_3;
	} else if (phi2 <= 1.02e+27) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_0))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -3.35e-6)
		tmp = t_3;
	elseif (phi2 <= 1.02e+27)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.35e-6], t$95$3, If[LessEqual[phi2, 1.02e+27], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.35e-6 or 1.0199999999999999e27 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -3.35e-6 < phi2 < 1.0199999999999999e27

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := t\_2 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1 \leq 0.03:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\frac{1}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right), t\_0, 0.5 - \cos \left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)}^{-0.5}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), t\_0, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3
         (+ t_2 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)) 0.03)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (*
      (*
       (atan2
        (/
         1.0
         (pow
          (fma
           (fma (cos (- lambda1 lambda2)) -0.5 0.5)
           t_0
           (- 0.5 (* (cos (- phi1 phi2)) 0.5)))
          -0.5))
        (sqrt
         (fma
          (fma (cos (- lambda2 lambda1)) 0.5 -0.5)
          t_0
          (fma (cos (- phi2 phi1)) 0.5 0.5))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = t_2 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.03) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = (atan2((1.0 / pow(fma(fma(cos((lambda1 - lambda2)), -0.5, 0.5), t_0, (0.5 - (cos((phi1 - phi2)) * 0.5))), -0.5)), sqrt(fma(fma(cos((lambda2 - lambda1)), 0.5, -0.5), t_0, fma(cos((phi2 - phi1)), 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(t_2 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.03)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(Float64(atan(Float64(1.0 / (fma(fma(cos(Float64(lambda1 - lambda2)), -0.5, 0.5), t_0, Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) * 0.5))) ^ -0.5)), sqrt(fma(fma(cos(Float64(lambda2 - lambda1)), 0.5, -0.5), t_0, fma(cos(Float64(phi2 - phi1)), 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.03], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(1.0 / N[Power[N[(N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$0 + N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * t$95$0 + N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.029999999999999999

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.029999999999999999 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. pow1/2 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)\right)}^{\frac{1}{2}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. metadata-eval N/A

         \[\leadsto \left(\tan^{-1}_* \frac{{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. pow-neg N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\frac{1}{{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)\right)}^{\frac{-1}{2}}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\frac{1}{{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)\right)}^{\frac{-1}{2}}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-unsound-pow.f64 57.9

         \[\leadsto \left(\tan^{-1}_* \frac{\frac{1}{\color{blue}{{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right)}^{-0.5}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   9. Applied rewrites 57.9%

      \[\leadsto \left(\tan^{-1}_* \frac{\color{blue}{\frac{1}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), -0.5, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)}^{-0.5}}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := t\_1 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := \cos \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;t\_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_4\right) \cdot t\_4 \leq 0.04:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, \mathsf{fma}\left(-0.5, t\_0, 0.5\right), 0.5 - t\_5 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_2, \mathsf{fma}\left(t\_5, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3
         (+ t_1 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (cos (- phi2 phi1))))
   (if (<= (+ t_1 (* (* (* (cos phi1) (cos phi2)) t_4) t_4)) 0.04)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (*
      (*
       (atan2
        (sqrt (fma t_2 (fma -0.5 t_0 0.5) (- 0.5 (* t_5 0.5))))
        (sqrt (fma (fma t_0 0.5 -0.5) t_2 (fma t_5 0.5 0.5))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = t_1 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = cos((phi2 - phi1));
	double tmp;
	if ((t_1 + (((cos(phi1) * cos(phi2)) * t_4) * t_4)) <= 0.04) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = (atan2(sqrt(fma(t_2, fma(-0.5, t_0, 0.5), (0.5 - (t_5 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_2, fma(t_5, 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = Float64(t_1 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = cos(Float64(phi2 - phi1))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_4) * t_4)) <= 0.04)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_2, fma(-0.5, t_0, 0.5), Float64(0.5 - Float64(t_5 * 0.5)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_2, fma(t_5, 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], 0.04], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$2 * N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] + N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$2 + N[(t$95$5 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0400000000000000008

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.0400000000000000008 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 61.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -2.35 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 5.6 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{e^{\log \left(1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -2.35e-7)
     t_2
     (if (<= phi2 5.6e-31)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
          (exp
           (*
            (log
             (-
              1.0
              (fma
               (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
               (cos phi1)
               (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))))
            0.5)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi2 <= -2.35e-7) {
		tmp = t_2;
	} else if (phi2 <= 5.6e-31) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))), exp((log((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1)))))))) * 0.5))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi2 <= -2.35e-7)
		tmp = t_2;
	elseif (phi2 <= 5.6e-31)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))), exp(Float64(log(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))) * 0.5)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.35e-7], t$95$2, If[LessEqual[phi2, 5.6e-31], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.35e-7 or 5.5999999999999998e-31 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -2.35e-7 < phi2 < 5.5999999999999998e-31

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{e^{\log \left(1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\right) \cdot 0.5}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \mathsf{fma}\left(-0.5, t\_0, 0.5\right)\\ t_2 := \cos \left(\phi_2 - \phi_1\right)\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{+62}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, t\_1, 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot t\_0 - 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 160:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{e^{\log \left(1 - \mathsf{fma}\left(t\_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot t\_0\right)\right) - 0.5 \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_3, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (fma -0.5 t_0 0.5))
        (t_2 (cos (- phi2 phi1)))
        (t_3 (* (cos phi2) (cos phi1))))
   (if (<= phi2 -3.2e+62)
     (*
      (*
       (atan2
        (sqrt (fma t_3 t_1 (- 0.5 (* t_2 0.5))))
        (sqrt (+ 0.5 (fma 0.5 (cos phi2) (* (cos phi2) (- (* 0.5 t_0) 0.5))))))
       2.0)
      R)
     (if (<= phi2 160.0)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (exp
           (*
            (log
             (-
              1.0
              (fma t_1 (cos phi1) (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))))
            0.5)))))
       (*
        (*
         (atan2
          (sqrt
           (- (+ 0.5 (* (cos phi2) (+ 0.5 (* -0.5 t_0)))) (* 0.5 (cos phi2))))
          (sqrt (fma (fma t_0 0.5 -0.5) t_3 (fma t_2 0.5 0.5))))
         2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = fma(-0.5, t_0, 0.5);
	double t_2 = cos((phi2 - phi1));
	double t_3 = cos(phi2) * cos(phi1);
	double tmp;
	if (phi2 <= -3.2e+62) {
		tmp = (atan2(sqrt(fma(t_3, t_1, (0.5 - (t_2 * 0.5)))), sqrt((0.5 + fma(0.5, cos(phi2), (cos(phi2) * ((0.5 * t_0) - 0.5)))))) * 2.0) * R;
	} else if (phi2 <= 160.0) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), exp((log((1.0 - fma(t_1, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1)))))))) * 0.5))));
	} else {
		tmp = (atan2(sqrt(((0.5 + (cos(phi2) * (0.5 + (-0.5 * t_0)))) - (0.5 * cos(phi2)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = fma(-0.5, t_0, 0.5)
	t_2 = cos(Float64(phi2 - phi1))
	t_3 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (phi2 <= -3.2e+62)
		tmp = Float64(Float64(atan(sqrt(fma(t_3, t_1, Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(Float64(0.5 + fma(0.5, cos(phi2), Float64(cos(phi2) * Float64(Float64(0.5 * t_0) - 0.5)))))) * 2.0) * R);
	elseif (phi2 <= 160.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), exp(Float64(log(Float64(1.0 - fma(t_1, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))) * 0.5)))));
	else
		tmp = Float64(Float64(atan(sqrt(Float64(Float64(0.5 + Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * t_0)))) - Float64(0.5 * cos(phi2)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * t$95$0 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.2e+62], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * t$95$1 + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(0.5 * t$95$0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 160.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(1.0 - N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 + N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$3 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.19999999999999984e62

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}}} \cdot 2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\left(\frac{1}{2} \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}}} \cdot 2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \phi_2}, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower--.f64 44.2

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right) - 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   10. Applied rewrites 44.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\color{blue}{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right) - 0.5\right)\right)}}} \cdot 2\right) \cdot R \]

   if -3.19999999999999984e62 < phi2 < 160

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{e^{\log \left(1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\right) \cdot 0.5}}}\right) \]

   if 160 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\color{blue}{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \color{blue}{\frac{1}{2} \cdot \cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \color{blue}{\frac{1}{2}} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \color{blue}{\cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      10. lower-cos.f64 42.8

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - 0.5 \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   10. Applied rewrites 42.8%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 + \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - 0.5 \cdot \cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 60.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \mathsf{fma}\left(-0.5, t\_0, 0.5\right)\\ t_2 := \cos \left(\phi_2 - \phi_1\right)\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{+62}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, t\_1, 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot t\_0 - 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 160:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot t\_0\right)\right) - 0.5 \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.5, -0.5\right), t\_3, \mathsf{fma}\left(t\_2, 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (fma -0.5 t_0 0.5))
        (t_2 (cos (- phi2 phi1)))
        (t_3 (* (cos phi2) (cos phi1))))
   (if (<= phi2 -3.2e+62)
     (*
      (*
       (atan2
        (sqrt (fma t_3 t_1 (- 0.5 (* t_2 0.5))))
        (sqrt (+ 0.5 (fma 0.5 (cos phi2) (* (cos phi2) (- (* 0.5 t_0) 0.5))))))
       2.0)
      R)
     (if (<= phi2 160.0)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             t_1
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       (*
        (*
         (atan2
          (sqrt
           (- (+ 0.5 (* (cos phi2) (+ 0.5 (* -0.5 t_0)))) (* 0.5 (cos phi2))))
          (sqrt (fma (fma t_0 0.5 -0.5) t_3 (fma t_2 0.5 0.5))))
         2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = fma(-0.5, t_0, 0.5);
	double t_2 = cos((phi2 - phi1));
	double t_3 = cos(phi2) * cos(phi1);
	double tmp;
	if (phi2 <= -3.2e+62) {
		tmp = (atan2(sqrt(fma(t_3, t_1, (0.5 - (t_2 * 0.5)))), sqrt((0.5 + fma(0.5, cos(phi2), (cos(phi2) * ((0.5 * t_0) - 0.5)))))) * 2.0) * R;
	} else if (phi2 <= 160.0) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(t_1, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = (atan2(sqrt(((0.5 + (cos(phi2) * (0.5 + (-0.5 * t_0)))) - (0.5 * cos(phi2)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = fma(-0.5, t_0, 0.5)
	t_2 = cos(Float64(phi2 - phi1))
	t_3 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (phi2 <= -3.2e+62)
		tmp = Float64(Float64(atan(sqrt(fma(t_3, t_1, Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(Float64(0.5 + fma(0.5, cos(phi2), Float64(cos(phi2) * Float64(Float64(0.5 * t_0) - 0.5)))))) * 2.0) * R);
	elseif (phi2 <= 160.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = Float64(Float64(atan(sqrt(Float64(Float64(0.5 + Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * t_0)))) - Float64(0.5 * cos(phi2)))), sqrt(fma(fma(t_0, 0.5, -0.5), t_3, fma(t_2, 0.5, 0.5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * t$95$0 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.2e+62], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * t$95$1 + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(0.5 * t$95$0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 160.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 + N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * t$95$3 + N[(t$95$2 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.19999999999999984e62

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}}} \cdot 2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\left(\frac{1}{2} \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}}} \cdot 2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \phi_2}, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower--.f64 44.2

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right) - 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   10. Applied rewrites 44.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\color{blue}{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right) - 0.5\right)\right)}}} \cdot 2\right) \cdot R \]

   if -3.19999999999999984e62 < phi2 < 160

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if 160 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\color{blue}{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \color{blue}{\frac{1}{2} \cdot \cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \color{blue}{\frac{1}{2}} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - \frac{1}{2} \cdot \color{blue}{\cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}, \frac{-1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), \frac{1}{2}, \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      10. lower-cos.f64 42.8

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - 0.5 \cdot \cos \phi_2}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   10. Applied rewrites 42.8%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 + \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) - 0.5 \cdot \cos \phi_2}}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 60.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \mathsf{fma}\left(-0.5, t\_0, 0.5\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, t\_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot t\_0 - 0.5\right)\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 160:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (fma -0.5 t_0 0.5))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (* (cos phi2) (cos phi1))
              t_1
              (- 0.5 (* (cos (- phi2 phi1)) 0.5))))
            (sqrt
             (+ 0.5 (fma 0.5 (cos phi2) (* (cos phi2) (- (* 0.5 t_0) 0.5))))))
           2.0)
          R)))
   (if (<= phi2 -3.2e+62)
     t_2
     (if (<= phi2 160.0)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             t_1
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = fma(-0.5, t_0, 0.5);
	double t_2 = (atan2(sqrt(fma((cos(phi2) * cos(phi1)), t_1, (0.5 - (cos((phi2 - phi1)) * 0.5)))), sqrt((0.5 + fma(0.5, cos(phi2), (cos(phi2) * ((0.5 * t_0) - 0.5)))))) * 2.0) * R;
	double tmp;
	if (phi2 <= -3.2e+62) {
		tmp = t_2;
	} else if (phi2 <= 160.0) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(t_1, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = fma(-0.5, t_0, 0.5)
	t_2 = Float64(Float64(atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), t_1, Float64(0.5 - Float64(cos(Float64(phi2 - phi1)) * 0.5)))), sqrt(Float64(0.5 + fma(0.5, cos(phi2), Float64(cos(phi2) * Float64(Float64(0.5 * t_0) - 0.5)))))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -3.2e+62)
		tmp = t_2;
	elseif (phi2 <= 160.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * t$95$0 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(0.5 - N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(0.5 * t$95$0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -3.2e+62], t$95$2, If[LessEqual[phi2, 160.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.19999999999999984e62 or 160 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), 0.5, 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}}} \cdot 2\right) \cdot R \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \color{blue}{\left(\frac{1}{2} \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}}} \cdot 2\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \phi_2}, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      6. lower--.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      8. lower-cos.f64 N/A

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}{\sqrt{\frac{1}{2} + \mathsf{fma}\left(\frac{1}{2}, \cos \phi_2, \cos \phi_2 \cdot \left(\frac{1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) - \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

      9. lower--.f64 44.2

         \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right) - 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   10. Applied rewrites 44.2%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\color{blue}{0.5 + \mathsf{fma}\left(0.5, \cos \phi_2, \cos \phi_2 \cdot \left(0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right) - 0.5\right)\right)}}} \cdot 2\right) \cdot R \]

   if -3.19999999999999984e62 < phi2 < 160

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 48.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(t\_0, -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{-19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 5.6 \cdot 10^{-119}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - t\_0 \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (fma (cos phi1) (fma t_0 -0.5 0.5) (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi1 -2.3e-19)
     t_2
     (if (<= phi1 5.6e-119)
       (*
        (atan2
         (sqrt
          (fma
           (cos phi2)
           (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
           (pow (sin (* -0.5 phi2)) 2.0)))
         (sqrt
          (- 1.0 (fma (- 0.5 (* t_0 0.5)) (cos phi1) (* (* 0.25 phi1) phi1)))))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = fma(cos(phi1), fma(t_0, -0.5, 0.5), pow(sin((0.5 * phi1)), 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi1 <= -2.3e-19) {
		tmp = t_2;
	} else if (phi1 <= 5.6e-119) {
		tmp = atan2(sqrt(fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - fma((0.5 - (t_0 * 0.5)), cos(phi1), ((0.25 * phi1) * phi1))))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = fma(cos(phi1), fma(t_0, -0.5, 0.5), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi1 <= -2.3e-19)
		tmp = t_2;
	elseif (phi1 <= 5.6e-119)
		tmp = Float64(atan(sqrt(fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(t_0 * 0.5)), cos(phi1), Float64(Float64(0.25 * phi1) * phi1))))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(t$95$0 * -0.5 + 0.5), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.3e-19], t$95$2, If[LessEqual[phi1, 5.6e-119], N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.2999999999999998e-19 or 5.6e-119 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{-1}{2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. lower-fma.f64 44.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 44.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{-1}{2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       12. lower-fma.f64 44.4

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 44.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   if -2.2999999999999998e-19 < phi1 < 5.6e-119

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.6%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 23.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 23.4%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 46.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (fma
      (cos phi1)
      (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
      (pow (sin (* 0.5 phi1)) 2.0)))
    (sqrt
     (-
      1.0
      (fma
       (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
       (cos phi1)
       (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

9. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
10. Add Preprocessing

Alternative 29: 46.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_0, 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - t\_0 \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (fma
          (fma -0.5 t_0 0.5)
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
   (if (<= (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))) 5e-8)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi1)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (* 0.25 (pow phi1 2.0))))
        (sqrt
         (-
          1.0
          (fma (- 0.5 (* t_0 0.5)) (cos phi1) (* (* 0.25 phi1) phi1)))))))
     (* (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) 2.0) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = fma(fma(-0.5, t_0, 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double tmp;
	if ((2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3)))) <= 5e-8) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), (0.25 * pow(phi1, 2.0)))), sqrt((1.0 - fma((0.5 - (t_0 * 0.5)), cos(phi1), ((0.25 * phi1) * phi1))))));
	} else {
		tmp = (atan2(sqrt(t_2), sqrt((1.0 - t_2))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = fma(fma(-0.5, t_0, 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))) <= 5e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), Float64(0.25 * (phi1 ^ 2.0)))), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(t_0 * 0.5)), cos(phi1), Float64(Float64(0.25 * phi1) * phi1)))))));
	else
		tmp = Float64(Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-8], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.25 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 4.9999999999999998e-8

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 22.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}}\right) \]

   if 4.9999999999999998e-8 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 33.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_1 := \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 5.6 \cdot 10^{-119}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_1}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi2)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_1
         (fma
          (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))
          (cos phi1)
          (* (* 0.25 phi1) phi1)))
        (t_2 (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_0))) (* 2.0 R))))
   (if (<= phi1 -3.1e-52)
     t_2
     (if (<= phi1 5.6e-119)
       (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_1))) (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0));
	double t_1 = fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), cos(phi1), ((0.25 * phi1) * phi1));
	double t_2 = atan2(sqrt(t_1), sqrt((1.0 - t_0))) * (2.0 * R);
	double tmp;
	if (phi1 <= -3.1e-52) {
		tmp = t_2;
	} else if (phi1 <= 5.6e-119) {
		tmp = atan2(sqrt(t_0), sqrt((1.0 - t_1))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_1 = fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), cos(phi1), Float64(Float64(0.25 * phi1) * phi1))
	t_2 = Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_0))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -3.1e-52)
		tmp = t_2;
	elseif (phi1 <= 5.6e-119)
		tmp = Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_1))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.1e-52], t$95$2, If[LessEqual[phi1, 5.6e-119], N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.0999999999999999e-52 or 5.6e-119 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.6%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 29.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 29.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   if -3.0999999999999999e-52 < phi1 < 5.6e-119

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.6%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 23.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 23.4%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 33.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{-19}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 5.6 \cdot 10^{-119}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t\_1, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))
          (cos phi1)
          (* (* 0.25 phi1) phi1)))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2
         (*
          (atan2
           (sqrt t_0)
           (sqrt (- 1.0 (fma (cos phi1) t_1 (pow (sin (* 0.5 phi1)) 2.0)))))
          (* 2.0 R))))
   (if (<= phi1 -2.3e-19)
     t_2
     (if (<= phi1 5.6e-119)
       (*
        (atan2
         (sqrt (fma (cos phi2) t_1 (pow (sin (* -0.5 phi2)) 2.0)))
         (sqrt (- 1.0 t_0)))
        (* 2.0 R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), cos(phi1), ((0.25 * phi1) * phi1));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = atan2(sqrt(t_0), sqrt((1.0 - fma(cos(phi1), t_1, pow(sin((0.5 * phi1)), 2.0))))) * (2.0 * R);
	double tmp;
	if (phi1 <= -2.3e-19) {
		tmp = t_2;
	} else if (phi1 <= 5.6e-119) {
		tmp = atan2(sqrt(fma(cos(phi2), t_1, pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - t_0))) * (2.0 * R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), cos(phi1), Float64(Float64(0.25 * phi1) * phi1))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - fma(cos(phi1), t_1, (sin(Float64(0.5 * phi1)) ^ 2.0))))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -2.3e-19)
		tmp = t_2;
	elseif (phi1 <= 5.6e-119)
		tmp = Float64(atan(sqrt(fma(cos(phi2), t_1, (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - t_0))) * Float64(2.0 * R));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.3e-19], t$95$2, If[LessEqual[phi1, 5.6e-119], N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.2999999999999998e-19 or 5.6e-119 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.6%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 29.1

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 29.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   if -2.2999999999999998e-19 < phi1 < 5.6e-119

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.6%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 23.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 23.4%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 27.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{0.5}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (fma
      (cos phi1)
      (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
      (* 0.25 (pow phi1 2.0))))
    (pow
     (-
      1.0
      (fma
       (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))
       (cos phi1)
       (* (* 0.25 phi1) phi1)))
     0.5)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), (0.25 * pow(phi1, 2.0)))), pow((1.0 - fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), cos(phi1), ((0.25 * phi1) * phi1))), 0.5)));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), Float64(0.25 * (phi1 ^ 2.0)))), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), cos(phi1), Float64(Float64(0.25 * phi1) * phi1))) ^ 0.5))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.25 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 22.0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 22.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 27.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{0.5}}}\right) \]
16. Add Preprocessing

Alternative 33: 27.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.25 \cdot \phi_1\right) \cdot \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_3 := \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, t\_0\right)\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \leq 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_3}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5, \cos \phi_1, t\_0\right)\right)}^{0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* 0.25 phi1) phi1))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (t_3 (fma (- 0.5 (* (cos (- lambda2 lambda1)) 0.5)) (cos phi1) t_0)))
   (if (<= (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))) 1e-9)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi1)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (* 0.25 (pow phi1 2.0))))
        (sqrt (- 1.0 t_3)))))
     (*
      (atan2
       (sqrt t_3)
       (pow
        (- 1.0 (fma (- 0.5 (* (cos (- lambda1 lambda2)) 0.5)) (cos phi1) t_0))
        0.5))
      (* 2.0 R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (0.25 * phi1) * phi1;
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double t_3 = fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), cos(phi1), t_0);
	double tmp;
	if ((2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2)))) <= 1e-9) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), (0.25 * pow(phi1, 2.0)))), sqrt((1.0 - t_3))));
	} else {
		tmp = atan2(sqrt(t_3), pow((1.0 - fma((0.5 - (cos((lambda1 - lambda2)) * 0.5)), cos(phi1), t_0)), 0.5)) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(0.25 * phi1) * phi1)
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	t_3 = fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), cos(phi1), t_0)
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))) <= 1e-9)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), Float64(0.25 * (phi1 ^ 2.0)))), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(atan(sqrt(t_3), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5)), cos(phi1), t_0)) ^ 0.5)) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-9], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.25 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 1.00000000000000006e-9

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 22.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}}\right) \]

   if 1.00000000000000006e-9 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 22.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 22.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.6%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]
   16. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}} \cdot \left(2 \cdot R\right) \]

       2. pow1/2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{\frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

       3. lower-pow.f64 24.8

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]

   17. Applied rewrites 24.8%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 34: 24.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.25 \cdot \phi_1\right) \cdot \phi_1\\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, t\_0\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5, \cos \phi_1, t\_0\right)\right)}^{0.5}} \cdot \left(2 \cdot R\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* 0.25 phi1) phi1)))
   (*
    (atan2
     (sqrt (fma (- 0.5 (* (cos (- lambda2 lambda1)) 0.5)) (cos phi1) t_0))
     (pow
      (- 1.0 (fma (- 0.5 (* (cos (- lambda1 lambda2)) 0.5)) (cos phi1) t_0))
      0.5))
    (* 2.0 R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (0.25 * phi1) * phi1;
	return atan2(sqrt(fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), cos(phi1), t_0)), pow((1.0 - fma((0.5 - (cos((lambda1 - lambda2)) * 0.5)), cos(phi1), t_0)), 0.5)) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(0.25 * phi1) * phi1)
	return Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), cos(phi1), t_0)), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(lambda1 - lambda2)) * 0.5)), cos(phi1), t_0)) ^ 0.5)) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]}, N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 22.0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 22.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.6%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]
16. Step-by-step derivation

    1. lift-sqrt.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}} \cdot \left(2 \cdot R\right) \]

    2. pow1/2 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{\frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

    3. lower-pow.f64 24.8

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]

17. Applied rewrites 24.8%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_1 - \lambda_2\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]
18. Add Preprocessing

Alternative 35: 19.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)\\ \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))
          (+ 1.0 (* -0.5 (pow phi1 2.0)))
          (* (* 0.25 phi1) phi1))))
   (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) (* 2.0 R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (cos((lambda2 - lambda1)) * 0.5)), (1.0 + (-0.5 * pow(phi1, 2.0))), ((0.25 * phi1) * phi1));
	return atan2(sqrt(t_0), sqrt((1.0 - t_0))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)), Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))), Float64(Float64(0.25 * phi1) * phi1))
	return Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.25 * phi1), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]}, N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 22.0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 22.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.6%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

16. Taylor expanded in phi1 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
17. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

    3. lower-pow.f64 19.6

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

18. Applied rewrites 19.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_1, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

19. Taylor expanded in phi1 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
20. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\frac{1}{4} \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

    3. lower-pow.f64 19.6

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

21. Applied rewrites 19.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(0.25 \cdot \phi_1\right) \cdot \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
22. Add Preprocessing

Reproduce

herbie shell --seed 2025155 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))